DYNAMICS AND RIGIDITY SIMION FILIP Spring 2018 Abstract. Course Notes for Math 253, Spring 2018, Harvard. Please let me know of any inaccuracies. Contents List of Corrections2 1. Introduction3 1.1. Abelian Rigidity3 1.2. Unipotent Rigidity5 1.3. Non-Abelian Stiffness8 1.4. Higher-rank lattices and Super-Rigidity9 2. Ergodic Theorems 11 2.1. Birkhoff and Kingman Ergodic Theorems 11 2.2. L2 ergodic theorems 16 2.3. Coverings and Maximal Theorems 17 2.4. Filling Scheme 20 2.5. Oseledets Theorem 20 2.6. Examples 23 3. Algebraic Groups and Ergodic Theory 25 3.1. Algebraic Groups: the basics 26 3.2. Algebraic Actions of Algebraic Groups 29 3.3. Chevalley’s theorems 31 3.4. A bit of structure theory of algebraic groups 33 3.5. Lie Algebras 37 3.6. Real semisimple Lie algebras and groups 40 4. Unitary Representations 44 4.1. Definitions 44 4.2. Mautner Phenomenon and Howe–Moore theorem 47 4.3. Property (T) 49 Revised April 24, 2018 . 1 2 SIMION FILIP 4.4. Amenability 55 4.5. Aside: Unitary representations of abelian groups 60 5. Super-Rigidity and Normal Subgroup Theorem 60 5.1. Super-Rigidity 61 5.2. Proof of step 1 of super-rigidity 63 5.3. Proof of step 2 of super-rigidity 65 5.4. Proof of super-rigidity 68 5.5. Arithmeticity of lattices 69 5.6. Margulis Normal Subgroup Theorem 69 6. Unipotent rigidity 71 6.1. Horocycle flow rigidity 71 6.2. Non-divergence of unipotent orbits 75 6.3. Oppenheim conjecture 75 7. Entropy 79 7.1. Shannon entropy 80 7.2. Kolmogorov–Sinai entropy 83 7.3. Conditional measures and Entropy 88 7.4. Local properties of entropy 89 8. Abelian rigidity 90 8.1. Furstenberg’s ×2 × 3 Theorem 90 8.2. Lyons–Rudolph theorem 92 8.3. S-arithmetic point of view 94 9. Exercises 100 9.1. General Dynamics 100 9.2. Algebraic Groups and Measure Theory 100 9.3. Entropy 102 9.4. Measure and Topological Rigidity 102 9.5. Other places 103 Appendix A. Transfer operator 104 References 104 List of Corrections Fill in details. 32 Add discussion of cohomology, fixed points for affine actions. Give an application of fixed points, e.g. acting on hyperbolic spaces with fixed points. 55 I need to see where I’m using that Γ has non-trivial image, when H has no compact factors. 68 DYNAMICS AND RIGIDITY 3 1. Introduction The course will be concerned with maps X → X from a space to itself. The simplest case is when the maps generate a group like Z or R, but our context will often have a larger group of maps, such as a Lie group (perhaps p-adic) with SL2 R a representative example. Here are two basic examples to keep in mind, in both cases we’ll have transformations of the circle T1 := R/Z. For the first, fix an irrational α and consider the translation Tα(x) := x + α mod 1 This action is isometric for the natural metric (see Exercise 9.1.1). All points behave uniformly for this transformation: every orbit is dense, in fact equidistributed in an appropriate sense. At the opposite extreme is the doubling transformation M2(x) := 2x mod 1 This action is uniformly expanding on the circle, and the orbit of a point x is determined by its base 2 expansion. The distribution properties of an orbit can be as chaotic as those of a random coin flip. The goal of the course will be to explore situations when one can make uniform conclusions as in the first example, but when the setup is closer to that of the second example. A key tool is to study invariant probability measures on X. Given a transformation T : X → X and a measure µ on X, its push-forward is defined by T∗µ(A) := µ(T −1A). This action extends the transformation to the space of all (say probability) measures, which is more flexible. The key point of most statements below is that they hold for all orbits of points x ∈ X, and not just for almost all. General ergodic theory can yield statements that hold for almost all points, in a natural measure-theoretic sense. One speaks of rigidity if there is a statement that holds for all orbits, and the possibilities are easily enumerated. 1.1. Abelian Rigidity The results in this category are less definitive and more than what is currently known is expected to be true. The first instance of the phenomena discussed is the following result of Furstenberg. 1.1.1. Theorem (Furstenberg [Fur67], topological ×2, ×3 rigidity). D E Consider the multiplicative semigroup Γ := 2a · 3b : a, b ∈ N acting on the circle T1 := R/Z, by descending the multiplicative action from R. Then the Γ-orbit of any irrational x ∈ T1 is dense. 4 SIMION FILIP The following measure rigidity analogue is still open. 1.1.2. Conjecture. For the same action as above, the only ergodic Γ- invariant probability measures are either atomic, or Lebesgue measure. Rudolph’s theorem says that the only possible exceptions to the above conjecture must have zero entropy for the action of any element of the semigroup. One interpretation of the above statements is that for an irrational, there should be no relation between the digits of the expansion in base 2 and base 3. A seemingly unrelated statement is the following question. 1.1.3. Conjecture (Littlewood). Let α, β ∈ R be irrational numbers, and for a real number γ denote by kγk the distance to the nearest integer. Then lim inf n · knαk · knβk = 0 n→∞ It turns out that this conjecture is equivalent to a statement about dynamics on homogeneous spaces. Here is the connection. SL ( ). 1.1.4. Proposition. Consider in the space of lattices 3 R SL3(Z) the lattice Λα,β spanned by the vectors 1 0 0 α , 1 , 0 β 0 1 Let A be the group of positive diagonal matrices of determinant 1. Then the orbit A · Λα,β is not contained in a compact set if and only if the pair (α, β) satisfies Littlewood’s conjecture. 3 Proof. For a vector v = (v1, v2, v3) ∈ R , the product of its coordinates v1 · v2 · v3 is invariant under A. By Mahler’s compactness criterion, a set of lattices is not contained in a compact set if and only if there is a sequence of non-zero vectors in the lattices whose norms go to zero (for a fixed ambient metric). For the case at hand, this is equivalent to there being a sequence of + elements ai ∈ A and vectors vi ∈ ai · Λα,β \{0} such that |vi| → 0, where |vi| denotes their norm for a fixed metric. By construction, the products of coordinates of vi goes to zero as −1 well. But vi and ai vi ∈ Λα,β have the same products of coordinates. Now an integral vector in Λα,β is of the form k1 k1α + k2 for ki ∈ Z k1β + k3 DYNAMICS AND RIGIDITY 5 The claim that there exist k1, k2, k3 such that the product of entries is arbitrarily small is equivalent to Littlewood’s conjecture. Unfortunately, the conjectured measure or topological rigidity results are not yet known in the generality needed. However, Einsiedler, Ka- tok, and Lindenstrauss proved that the set of possible exceptions to Littlewood’s conjecture has Hausdorff dimension 0. The methods that are currently available do give the following appli- cation. 1.1.5. Theorem (Lindenstrauss, Arithmetic Quantum Unique Ergod- icity). Suppose that Γ ⊂ SL2(R) is an arithmetic lattice. Let φi be a sequence of normalized eigenfunctions of the Laplace operator on the hyperbolic surface H2/Γ, with eigenvalues tending to infinity. 2 Then the sequence of probability measures |φi| dVol tends weakly to Lebesgue measure. The result is saying that no subsequence of eigenfunctions can concen- trate unevenly on subsets of the surface. Again, by general principles one knows (Shnirelman) that for a sequence of eigenfunctions of positive upper density the statement holds. The key point is to show it for any sequence of eigenfunctions. The relation between Theorem 1.1.5 and measure rigidity is via Hecke operators. The key point is the arithmetic structure of Γ, which says 1 that there exists a Q-algebraic group G such that Γ = G(Z) and SL2(R) = G(R). Then one can lift the situation to a measure on the space G(R) × G(Qp)/G(Z[1/p]) with a left A(R × Qp) − action where A is a split Cartan, i.e. a subgroup isomorphic to the diagonal matrices. The action of A(Qp) corresponds to the Hecke action. Unlike in the case of the Littlewood conjecture, one can show that the action of A does have some positive-entropy directions. This then suffices to apply the existing measure rigidity results. 1.2. Unipotent Rigidity A good introduction to unipotent measure rigidity are the notes of Eskin [Esk10]. The following result, whose statement involves only quadratic forms, was initially attacked by methods of analytic number theory and was 1 This is abusive, there isn’t really a G(Z) for a Q-algebraic group, but some finite index version of it 6 SIMION FILIP settled for sufficiently large dimensions. Raghunathan made the con- nection to unipotent flows, and Margulis settled the conjecture. 1.2.1. Theorem (Oppenheim Conjecture). Suppose that Q is an in- definite quadratic form on Rn with n ≥ 3. If Q is not proportional to a rational form, then for any ε > 0 there exists v ∈ Zn such that 0 < |Q(v)| < ε Equivalently, the values Q(Zn) accumulate to 0.